The kernel/initrd come from a regular OCI container image — there's no separate VM image format. Any OCI image containing /boot/vmlinux (and optionally /boot/initramfs.cpio.gz) works as a kernel image.
you don't have to source one manually — pullrun kernel install fetches a kata containers kernel into ~/.pullrun/kernels/vmlinux-<version>. One-time setup, then the daemon auto-discovers it.
If you run an ubuntu image with a kata kernel, do you have difficulty installing some packages? For example if you need linux-headers-$(uname -r) or linux-modules-extra-$(uname -r), it won't exist in canonical apt repos?
I don't think I've ever seen a kernel or initrd in a container image. The whole point of containers is not to duplicate the kernel, and you definitely don't need an initial ram disk since you already have a running operating system.